Which burst sizes guarantee a noise trade-off? Geometric bursts are the only ones, for a common burst law and for pairs of laws

Unrefereed research candidate. Many molecules in cells are made in bursts: one production event releases a random number of copies. A long-standing conjecture says feedback between species cannot make every species less noisy than constant-rate production at the same mean, with noise measured by the Fano factor (variance divided by mean). Anderson (2026, arXiv:2607.20823) proved this when burst sizes are geometric, found burst laws on {1,2} for which it fails, and asked which other burst laws still enforce a trade-off and how it changes away from geometric. This package answers both questions. Theorem 1: for a burst law shared by all species, a universal trade-off holds if and only if the law is positive geometric; every nongeometric law with finite second moment, even one with the same mean and variance as a geometric law, lets a two-species loop (one activation, one repression) push both Fano factors below baseline at low production intensity. Theorem 2: for two species with different burst laws, simultaneous suppression in the same loop is possible exactly when at least one law is nongeometric. Theorem 3: the aggregate trade-off fails by at most the regulatory rate range times the total-variation distance between each burst law and its equilibrium transform, which vanishes exactly for geometric laws; this bound uses lemmas of Anderson's preprint, stated with hypotheses. Exact examples: the law P(K=1)=2/3, P(K=4)=1/3 gives Fano factors 2 − (19/11520)ε2 + O(ε3); a geometric law paired with it gives coefficients −233/2949120 and at most −3.8057×10−4. Effect sizes: tiny, of order 10−6 in the Fano factor, at mean abundances below 0.1 molecules in the examples. The results are mathematical statements about a standard stochastic model, not predictions for particular genes. The archive contains the 12-page manuscript (PDF and LaTeX), exact standard-library checkers that do not rely on assert, numerical illustration tools with pinned dependencies, nine semantic negative controls, CI and SHA-256 sums. Assurance: written proofs and exact rational checks of finite-support identities and examples; internal AI-assisted review only (not peer review, independent reproduction or formal verification); Theorem 3 depends on an unrefereed preprint; priority bounded to the recorded search. Creator: Anonymous. Text and data CC0-1.0; code MIT.